Relation between Anderson Generating Functions and Weil Pairing

TL;DR

This paper offers a new, simpler interpretation of the Weil pairing for Drinfeld modules, connecting it with Moore determinants and Anderson generating functions, extending previous explicit formulas.

Contribution

It introduces an elementary, explicit formula for the Weil pairing using Moore determinants and Anderson generating functions, avoiding Anderson motives.

Findings

Derived a simple interpretation of the Weil pairing as a Moore determinant coefficient.

Connected the Weil operator with the remainder polynomial of Anderson generating functions.

Extended explicit formulas to higher ranks using Moore determinants.

Abstract

The existence of the Weil pairing for Drinfeld modules was proved by van~der~Heiden using the Anderson $t$-motive. Papikian's note provided the explicit formula for the rank-two Weil pairing that avoids Anderson motives. Following this approach, Katen extended the formula to higher ranks. As Papikian observed, this method is more elementary than the approach using Anderson motives, but it is less conceptual. This paper is devoted to a new insight into Katen's formula motivated by the Moore determinant coming from Hamahata's tensor product of Drinfeld modules and the basis of torsion modules found by Maurischat and Perkins. We investigate the Weil operator, establish its connection with the remainder polynomial of Anderson generating functions modulo a fixed polynomial $\f$, and finally derive an extremely simple interpretation: the value of the rank-$r$ Weil pairing is essentially the specific coefficient in the Moore determinant of certain Anderson generating functions.